{
"id": "1709.03957",
"version": "v1",
"published": "2017-09-12T17:14:59.000Z",
"updated": "2017-09-12T17:14:59.000Z",
"title": "Zeroes of the Swallowtail Integral",
"authors": [
"David Kaminski"
],
"comment": "10 pages, 2 figures",
"categories": [
"math.CA"
],
"abstract": "The swallowtail integral $S(x,y,z) = \\int_{-\\infty}^{\\infty} \\exp[i(u^5 + xu^3 + yu^2 + zu)] \\, du$ is one of the so-called canonical diffraction integrals used in optics, and plays a role in the uniform asymptotics of integrals exhibiting a confluence of up to four saddle points. In a 1984 paper by Connor, Curtis and Farrelly, the authors make a number of remarkable observations regarding the zeroes of $S(x,y,z)$, including that its zeroes occur on lines in $xyz$-space, and that the zeroes of $S(0,y,z)$ lie along the line $y = 0$. These assertions are based on numerical evidence and the asymptotics of $S(0,0,z)$. We examine these assertions more completely and provide additional detail on the structure of the zeroes of $S(x,y,z)$.",
"revisions": [
{
"version": "v1",
"updated": "2017-09-12T17:14:59.000Z"
}
],
"analyses": {
"subjects": [
"41A60",
"33E20"
],
"keywords": [
"swallowtail integral",
"uniform asymptotics",
"assertions",
"saddle points",
"canonical diffraction integrals"
],
"note": {
"typesetting": "TeX",
"pages": 10,
"language": "en",
"license": "arXiv",
"status": "editable"
}
}
}